Display control apparatus for drawing accurate lines

ABSTRACT

The development of a simplified arrangement for drawing arcuate lines on a display, such as a cathode ray tube face, by the successive specification of a sequence of displacement positions is explained, and apparatus is described for calculating the sequence of displacement positions by relative right-shifting and selective addition of a pair of initially calculated values, one for each of two co-ordinate axes. The apparatus includes a first register for each of the initial values and an accumulator for each axis, the values from the first registers being entered into each of the accumulators to form a new value. The resultant values are used to specify a drawing displacement for each axis and are recirculated to replace the values in the first registers, this cycle of events being repeated as required until the line is drawn. Where an arc extends beyond a quadrant bounded by the axes the signs of the values entered into the accumulators are changed as required for drawing an arc in the new quadrant. The change of sign is effectively accomplished by selective complementing of the values as they are entered into the accumulators.

United States Patent Hughes [45] Dec. 11, 1973 DISPLAY CONTROL APPARATUS FOR DRAWING ACCURATE LINES Primary Examinen-Carl D. Quarforth Assistant Examiner-J. M. Potenza Attorney-Frederick E. Hane X'ADDRILSS 'DECODER [57] ABSTRACT The development of a simplified arrangement for drawing arcuate lines on a display` such as a cathode ray tube face, by the successive specification of a se quence of displacement positions is explained, and apparatus is described for calculating the sequence of displacement positions by relative right-shifting and selective addition of a pair of initially calculated values, one foreach of two co-ordinate axes.

The apparatus includes a first register for each of the initial values and an accumulator for each axis, the values from the first registers being entered into each of the accumulators to form a new value. The resultant values are used to specify a drawing displacement for each axis and are recirculated to replace the values in the first registers, this cycle of events being repeated as required until the line is drawn.

Where an arc extends beyond a quadrant bounded by the axes the signs of the values entered into the accumulators are changed as required for drawing an arc in the new quadrant. The change of sign is effectively accomplished by selective complementing of the values as they are entered into the accumulators.

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sum a or 2 1 (4 ADDRESS 2 t vmnmss DECODER M DECODER l l H I2 A Vh AA A M m* ARC coNTRoL 25 UNH DISPLAY CONTROL APPARATUS FOR DRAWING ACCURATE LINES BACKGROUND OF THE INVENTION l. FIELD OF THE INVENTION The present invention relates to display control apparatus, and in particular to apparatus for drawing displays on the face of a cathode ray tube.

2.1DESCRIPTION OF THE PRIOR ART lt has previously been proposed to provide apparatus for drawing displays on the face of a cathode ray tube by the selective movement of the electron beam of the tube. Such apparatus has frequently been employed in conjunction with a computer and, for example, in performing a drawing operation the path to be traced out by movement of the beam is continuously computed in terms of increments of displacement of the beam in each of two orthogonal axes, the X and Y axes. In other cases the pattern to be drawn is computed and is stored in a storage device as a series of displacement values for each line to be drawn.

SUMMARY OF THE INVENTION According to ,the present invention apparatus for controlling drawing of arcuate lines on a display face includes means for initially deriving a pair of values representing for co-ordinate axes respectively the product of the radius of the required arc and the sine of a displacement angle through which the radius is to be moved by a predetermined incremental distance along the arc to be drawn, and the product of the radius and thecosine of the displacement angle 0, the pair of valuesbeing expressed in terms of said incremental distance in binary code notation having n denominations; means for temporarily registering-the pair of values; means for arithmetically accumulating independently for each axisirespectively the registered value for that axis right-shifted by n+1 denominations and the registered `:value for the other axis right-shifted by n/2 denominations; means for extracting from the accumulating means values from the n/2 most significant denominations; means for applying the extracted values respectively to specify for the coordinate axes a step of the arc to be drawn and for applying the extracted values to modify the registered values respectively of the same axis; and means for controlling repeated sequential operations of the accumulating means, the extracting means and the applying means to specify succeeding steps of arc drawing.

BRIEF DESCRIPTION OF THE DRAWINGS DESCRIPTION OF THE PREFERRED EMBODIMENT Before describing in detail apparatus for controlling the displayof arcuate lines on the face of a cathode ray tube arrangement, the requirements of such an arrangement will be briefly reviewed. In drawing lines on the display face of a cathode ray tube it will be realised that such lines are formed by the displacement of the electron beam of the tube, and that the apparent thickness of the lines drawn is dependent upon the resolution on the display face ofthe spot formed by the beam. It is also to be noted that because this spot has a nite size it is convenient to consi-der the beam displacement in drawing a line in terms of the spot resolution. The dimensions of figures to be drawn, and hence the degree of displacement required for the beam are also most easily specified in terms of increments of a size consistent with spot resolution.

Hence, the-magnitude ofthese increments sets then requirements for digital definition of spot position addresses required to specify beam movement over the display face.

British Patent application Ser. No. 15152/71 shows a straight line drawing arrangement in which beam position values are expressed in binary code notation and in which these values are decoded into analogue signals to control the positioning of the cathode ray tubel beam at the position whose address is specified in a pair of position registers, one for each of two orthogonal axes. In the case of arcuate lines, it will be appreciated that such lines may also be drawn by a similar beam positioning system. However, in drawing an arc, the incremental alteration to the address value in a position register for each drawing step must be such that a chord traced out by the beam movement does not deviate from the theoretical true arcuate path by a distance as great as the minimum resolution distance between two adjacent spot positions. In a practical case, the incremental distance is made equal to the spot resolution of the cathode ray tube, and the maximum difference in position between the chord actually traced out by beam movement and the true arcuate path is required to be less than one such increment. In the present apparatus, the radius of the arc is expressed in terms of these increments, and it will be appreciated that the maximum radial value for an arc which may be expressed in a binary code register is determined by the binary denominational capacity of that register, and is 2-l where n is the denominational capacity of the register.

It will be seen that the error between a drawn chord and the true arc will depend upon the notional angle through which the radius is moved for each step of drawing, and it can be shown that, for any circle, the maximum permissible error of one increment will occur for a notional angular radial movement da such that the sine of half the angle da is equal to the value (2R-1) /R, where R is the radius. As R becomes large in relation to the angular change, this limiting value may be taken as 2(2) /R which is always smaller than the theoretical case, and in practice this expression, too, may be further simplified with safety. As noted above the maximum value of R to be expressed in binary code notion is 2l. lf, however, this maximum value is notionally taken as 2", the critical angle may, with safety, be assumed to be 2(2) /2 for the case of maximum expressible radius.

Having arrived at a safe value for angular increment da in terms of R, it is now desirable to express increments of axial deflection which will not produce an unacceptable increment, also in terms of R. The two axes of displacement will be referred to as the X-axis and the Y-axis respectively, as shown in FIG. l. This gure illustrates the condition in which movement of a radius from a point a to a point b sweeps out an arc ab. From the figure it will be seen that the changes in the X and Y axis values, X and Y respectively result from movement of the radius through an angular increment di from an initial position at a notional angle d; with respect to the X-axis. The total angular movement is, of course, exaggerated for the sake of clarity. It will be appreciated that the change Y can be expressed as Y'( l-Cos ip) X' sin dawhere Y' is equal to R sin d and X' is equal to R cos 0. Similarly, the change X can be expressed as Y sin d) X(1-cos ib).

ln the solution of these equations to obtain values for the incremental changes in the X-and Y-axis position values it is required to include progressively higher terms of an expression for the incremental angle di. However, providedthat any error produced by arbitrarily terminating the progression cannot produce a beam displacement greater than one increment of resolution in drawing a complete circle, the solutions may be considerably simplified. Moreover, the terms to be used for calculation must, for convenience of manipulation in the apparatus to be described, be expressible by round binary coefficients.

lt can be shown that if the values of X and Y are assumed to be respectively YX2l2 and X+Y2/2, then for a complete circle the accumulated error for any quadrant increases with an increase in n (i.e. a decrease in di) and tends towards -F2/6, where F is a constant, defined by =Fl2 as becomes very small. Under these conditions, therefore, the total accumulated error in drawing a complete circle cannot exceed 4132/6, and if the requirement that this total error be less than one increment is considered in relation to the contents of a binary register containing incremental values, then this expression must be soluble to produce a value less than unity. Hence the value for F2 may not exceed 1.67 so that F cannot exceed 1.225. lf now the value for F is assumed to be 1, then the value for da becomes it, as previously noted, where nis the denominational capacity of the binary register. Using this expression l for d, the values for X and Y to be used in drawing an arc then become, respectively:

in terms of the values in the register. Thus assuming an initial starting position for the arc to be expressed as X- and Y-axis values in terms of the radius of the arc to be drawn relative to an initial notional angle lying within a predetermined quadrant as shown in FIG. 1, the values X Rcos0 and Y Rsinl) may be calculated and held in two multidenominatonal binary registers respectively. Taking as an example, the Y value, it is seen that the expression X/2 may be derived by dividing the starting value in the X register by that power of 2 equal to half the capacity of the register, and that this division is equivalent to a right-shift of the contents of the X- axis register by half its capacity. Moreover, the expression Y/2"Jfl is equivalent to a right-shift of n 1 places, where n is the Y-axis register capacity. Hence these two expressions may be derived by the appropriate rightsliifting of the contents of the registers and the values thus obtained are added or subtracted, as required by the respective equations above, to the values standing in the registers. This operation then provides the required modification of the positional values for controlling electron beam movement to draw an increment of the arc, and it will be seen that these modified values then become the starting point for drawing the next increment of the arc. Thus, this derivation and summing process continues until the arc is completely drawn.

FIG. 2 shows an arrangement of registers for carrying out this process of derivation and summing. A pair of arc value registers 31 and 32 are provided, respectively for the X and Y-axis values. ln the present case, each arc register 3l, 32 has eight binary denominational positions' and has input channels 33 and output channels 34. It will be understood that, in practice, the number of denominational positions is not limited to eight. An adder 35 is associated with the X-axis and a similar adder 36 is associated with the Y-axis, both adders having input and output channels 37 and 38 respectively, the output channels'38 being connected to adder output registers 39 and 40 for the X-and Y-axes respectively, The adder output registers are arranged to pass the values selectively in true or twos-complementary form to output channels 44.v An accumulator 7 is provided for the X-axis, and has a more significant portion 7a having, in the present case, four denominational positions and a less significant portion 7b having nine positions, the more significant half being termed the X register while the less significant half serves to hold X- increment remainder values. A similar accumulator 17 is provided for the Y-axis. Outputs from the X register 7a and the Y register 17a are taken over transfer paths 41 to X-and Y-position registers 5 and l5 respectively, The output channels 34 from the arc registers 3l and 32 are connected to the input channels 37 of the adders 35 and 36 respectively.

The adders 35 and 36 receive outputs from the arc registers 31, 32 of the same axis respectively from the channels 34, and the four least significant denominational positions of the adders 35, 36 also receive inputs from the X and Y registers 7a and 17a respectively. The values representing the sums of these input values are passed from the adders 35, 36 to the output channels 38, which are respectively connected to the output registers 39, 40. Output channels 44 from the output registers 39, 40 of each axis are respectively connected to the input channels 33 of the X-arc and Y-arc registers 31 and 32; and also to input channels 43 of the registers 7 and 17 of the same axis respectively. The same channels 44 from the registers 39 and 40 are also connected to input channels 42 of the registers 17 and 7 of the opposite axes respectively.

The connections to the channels 42 are respectively arranged so that values from the register 39, for example, are right-shifted by four denominational positions before being entered into the register 17. Thus, the more significant four denominations of the value in the output register 39 are entered into the Y register 17a, while the remaining less significant denominational values enter the most significant denominational positions of the remainder of the Y axis register 17. This disposition of the connections provides the required rightshift equal to half the capacity of the X-arc register. Similarly the values from the Y-axis output register 40 enter the X-axis register 7 with a n/2 denominational position shift applied to them.

The connections between the channels 44 and 43 are arranged to provide a right shift of n-l-l denominational positions in the transfer of values from the registers 39,40 into the X and Y remainder registers, so that the values from the adders 39, 40 actually enter the least significant positions of the register portions 7b and 17h. ln operation, the initial values for X=Rcos6 and Y=R sin are entered into the X-arc and Y-arc registers 3l and 32 respectively, which, it will be recalled, specify an initial starting point for an arc to be drawn.

Together with the values initially in the X and Y register portions '7a and 17a (which at this stage would be zero) the initial values from the X -arc and Y-arc registers 31, 32 are summed and passed to the output registers 39 and 40 respectively. The values from the registers 39 and 40 are first entered into the accumulators 17and 7, respectively associated with the opposite axis,

" with a right shift of n/2 positionseThus, at this point the values now in the register portions 7a and 17a represent incremental alterations to be applied to the current beam position equal to the first terms Y/2 and X/2 respectively of the expressions for X and Y given above, and are also, in a second phase of operation, transferred to the registers 7 and 17 respectively with the n+1 position right shift described (these values representing the second terms X/2n+1 and Y/2"*l respectively of the X and Y expressions). Thus, the transfer of the values from the registers 39, 40 updates the X -arc and Y-arc registers 31, 32 and completes the alteration to the values in the registers 7 and 17 to specify the end of the first arc drawing increment. This point on the drawing now becomes the starting point for the second increment, and the values in the output registers 39 and 40 are updated by addition of the values in the X and Y registers respectively to the X-arc and Y-arc registers 31, 32. The X and Y registers 7a and 17a are cleared, while the register portions 7b and l7b are allowed to continue to accumulate remainder increments. A new cycle of adding and transfer is then performed. These cyclic operations of entry, adding and transfer and clearing are repeated until the drawing of the arc is complete.

It will be seen from FIG. 1 that for the first quadrant shown, the X axis values are progressively increasing while the Y values are decreasing, and consideration will show that the progressive increase or decrease of values for the respective axes is dependent upon the quadrant which is currently being drawn. It will be re called that the output registers 39 and 40 (FIG. 2) include means for selectively complementing the values applied over the channels 44. The selection of true or complementary transfer of values is performed in accordance with the particular quadrant in which the arc to be drawn lies, so that the X and Y values are in positive or negative form as required for drawing the arc.

The X and Y values are transferred to the X- and Y- position registers 5 and 15 respectively over the lines 41,land this transfer path includes provision for adding the transferred values to those currently in the position registers 5, 15, so that the position values are progressively altered. The position registers are, in practice, coupled to the cathode ray tube beam deflection control circuitry to modify the beam position in accor dance with the changes registered by the X and Y registers 7a"and 17a respectively. It will be seen that the alteration of the values in the registers 7 and 17 for each increment of beam movement required effectively a number of adding steps; for example, one for updating the registers 39 and 40, others for implementing the addition of the first and second terms of the X and Y expressions asnoted above and for updating the X and Y-arc registers 31, 32. Thus the transfer of values from the registers 7a and 17a to the registers 5 and l5 respectively is synchronised with the cyclic sequence and takes place after appropriate ones of the adding steps.

An alternative arrangement for drawing arcuate lines by the same method is shown in FIG. 3, which will now be described in detail. The lines are drawn on the display face of a cathode ray tube l, by displacement of the electron beam of the tube 1 under control of signals applied to displacement electrodes 2 in conventional manner. The electrodes 2 receive signals respectively from an X-address decoder 3 and a Y-address decoder 4, which are conventional binary code digital to analogue converter networks. Binary coded position ad-Y dresses are applied to the decoders 3 and 4 by X and Y-position registers 5 and 15 respectively. X and Y registers 7a and 17a are provided and are coupled to the position registers 5, 15 by adders 6, 16 respectively for the X and Y axes, so that X and Y values are added to into the position registers 5, 15 to alter the beam position of the cathode ray tube l.

Values are transferred into the X and Y registers 7a and 17a from the four most significant denomination positions of accumulators 13 and 23 respectively, which each have thirteen positions in all. The accumulators 13 and 23 receive inputs from input gate groups 1l, 12, 21 and 22, which respectively deal with the expressions X/2 X/2"+; Y/2 and Y/2"+. Thus the gate group 1l has eight outputs connected respectively to the first to eighth positions of the accumulator 23; the gate group 12 has eight outputs connected respectively to the sixth to 13th positions of accumulator 13; the gate group 21 has eight outputs connected respectively to the first to eighth positions of accumulator 13 and the gate group 22 has eight outputs connected respectively to positions six to 13 of accumulator 23.

Gate groups 1l and l2 have their inputs connected in common through a selective complementer 10 to the respective outputs of an eight position X-arc register 8, while gate groups 2l and 22 have their inputs connected similarly through a complementer 20 to a Y-arc register 18. Values are initially entered into the X-arc and Y-arc registers 8 and 18 from a processor 30. Subsequent changes in the X-arc and Y-arc register values are applied through gating adding networks 14 and 24 respectively from the outputs of the four most significant positions of the accumulators 13 and 23 respectively, under control of an arc control unit 26, which also provides control signals to determine the operation of the gate groups 11, 12, 21 and 22, the adders 6, and 16, the complementers l0 and 20 and the X and Y registers 7a and 17a. The networks 14 and 24 are adding networks of which the X- and Y-arc registers 8, 18 also form output registers, and the networks include output gates to permit a sum value to pass into these output registers 8, 18. Thus, the networks 14, 24 are also connected so that the values in the registers 8, 18 also form second inputs respectively.

The X and Y registers 7, 17 have limit indicators 9 and 19 respectively associated with their output lines. The indicators 9 and 19 are accumulating comparators which also receive inputs from a radius register 25. The radius register stores a value representing the radius of the arc to be drawn, and the indicators 9 and 19 are each arranged to provide an indication to the control unit 26 when the accumulated displacement of the respective axis approaches within one increment of the arc radius, thus indicating the completion of the drawing of a quadrant of a circle.

It will be understood that because the indicators 9 and 19 are effective to indicate the completion of a quadrant, they may take other forms. For example, an alternative form of indicator could consist of an arrangement of gates associated with the stages of the X- or Y-arc register and arranged to indicate the approach of the value in the register to zero.

The radius value for the register 2S is obtained from a display store 28 which also provides indications of the initial beam position for entry into the position registers 5 and 15 and the notional starting angle 0 in the particular quadrant from which the arc is to be drawn, this latter information being passed to the processor 30. The processor 30 provides the quadrant information to permit the control unit 26 to select the operation of the complementers l and 20 so that values passed to the accumulators 13 and 23 are in true or complementary form to permit the displacement values to be added or subtracted in accordance with whether the axial values are increasing or decreasing in drawing the arc.

In operation, the initial beam position values are entered into the registers and l5 from the store 28, which also provides the initial notional angle 0 information to the processor 30 and the radius value to the radius register 2S. The radius value is also passed to the processor 30, which calculates the X and Y values, RcosH and Rsin0, and enters these values into the X- and Y-arc registers 8 and 18 respectively.

The first step of the displacement calculation is now performed under control of the unit 26, which conditions the gates 11 and 21 and the complementers 10, 20 to permit the values from the registers 8 and 18, in true or complementary form as required, to-be entered into the accumulators 23 and 21 of the opposite axes respectively with what is, with respect to the registers 7a and 17a, an effective four-position right shift, thus adding the first expressions for the X and Y values into the accumulators.

The next calculation step is performed in a rather similar way. The gates 11 and 21 are deconditoned and the gates l2 and 22 are conditioned. The complementers and 20 are re-selected as required and the values from the X -arc and Y-arc registers are entered into the accumulators 13 and 23 of the same axis respectively with an effective nine-position right shift to add the second expressions of the X and Y values into the accumulators 13 and 23.

The third step now follows in which the complete most significant four denominations of the X and Y values are transferred from the accumulators 13 and 23 into the X and Y registers 7a and 17a. At the same time the networks 14 and 24 are conditioned to permit these values to be fed back to the X-arc and Y-arc registers 8 and 18 to update the values ftherein. While the registers 7a and 17a are shown in the present embodiment, it will be realised that they are not required for a simple arc drawing arrangement, but do permit, in a practical case, the present arc drawing facility to be provided in association with other display arrangements. Hence, for a simple application the X and Y values may be transferred directly into the position register 5 and l5 respectively.

In the present case, however, the X and Y values are transferred from the registers 7a and 17a into the position registers through adders 6 and 16 respectively to alter the position register values and thus move the cathode ray tube beam to draw an increment of the arc. The X and Y values are also passed to the limit indcators 9 and 19. The transfer of the X and Y values may, of course, take place while a new incremental value is being formed in the accumulators 13 and 23 or this transfer may take place as a fourth step of the calculating cycle. In addition, the registers 7a and 17a are cleared before the formation of new X and Y values.

The above operations are repeated until arc drawing is terminated because an end point is reached, the end point being specified iri the stored information relating to the arc. Where the end of a quadrant is indicated by the indicators 9 or 19, the drawing of the arc continues as required, but the selection of the complementers l0, 20 is changed to suit the new conditions for increase or decrease of axial values.

lt will be appreciated that in the example shown, in which the indicators 9 and 19 are connected to the registers 7a and 17a, if the arc begins at some intermediate point within a quadrant, the initial value in the limiting indicators 9, 19 must be present in order that a true indication of end of quadrant is obtained. Thus, the indicators 9 and 19 are, in this case, primed with the initial X and Y position values which are entered into the arc registers. Thus, one of the indicators will then accumulate the successive displacement values to reach the radial value at the end of the quadrant. Where, as previously noted, the indicators 9, 19 are associated with the registers 8, 18, this difficulty does not arise.

The foregoing examples have assumed, for the purposes of clarity, that the value of R in terms of binary denominations necessary for its expression is constant and that the register denominational capacities are chosen to suit this value. However, consideration will show that account may be taken of values for R that involve significant and non-significant bits by adjust-ing the shifting requirements of the system.

In a system arranged for a maximum of n bits, for example, in which there are n significant bits then the values representing the expressions X /2 and Y/2 ,respectively, will require to be corrected by being left shifted by nn'/2 positions from those described, and this left-shifting may be effectively obtained by modifying the degree of right-shifting applied to the values during transfer to the accumulators. Thus, for example, the gating groups 11., l2, 21 and 22 may incorporate arrangements for providing the appropriate rightshifting in dependence upon the particular arcs to be drawn. In this case the appropriate degree of shifting could be specified or computed by the processor 30 before the drawing of each arc. The expressions X/2"* and Y/2"+l also require left shifting by n-n' positions to make a comparable correction. Where the n is odd, then these left-shift corrections are adjusted to become (n-n-l)/2 and nn-l respectively.

Other modes of operation are also possible by further modifications of the relative shifting requirements. For example, suppose that it is required to-draw the arc in multi-increments which are shorter than the previously noted maximum length. Then let the multi-increment be specified as m bits. Thenthe relative right shift for the X /2 and Y/2 expressions is n-m positions, and a further right shift of n'-m+l positions is required for the expressions X/Zn+1 and Y/2"+1.

In all cases, however, it will be apparent that the two expressions noted above are derived by differential shifting of the Rsin and Rcos@ values initially obtained and entered into the arc registers. These shifted values are accumulated in registers associated with the appropriate axes to form incremental values for altering the current beam position values for each step of arc drawing, and the progressively obtained incremental values are fed back to modify the initial values in readiness for the derivation of each succeeding new step.

I claim:

l. Apparatus for controlling drawing of arcuate lines on a display face, including means for initially deriving a pair of values representing for co-ordinate axes respectively the product of the radius of the required arc and the sine of a displacement angle through which the radius is to be moved by a predetermined incremental distance along the arc to be drawn and the product of the radius and the cosine of the displacement angle 0, the pair of values being expressed in terms of said incremental distance in binary code having n denominations; means for temporarily registering the pair of values in each axis independently; accumulating means for each axis respectively; means for entering values from the temporary registering means into the accumulating means to accumulate independently for each axis respectively the temporarily registered value .for the same axis right shifted by n l denominations and the temporarily registered value for `the other axis right shifted by n/2 denominations; means for extracting from the accumulating means for each axis respectively those values from the n/2 most significant denominations; means for applying the extracted values respectively to specify for the coordinate axes a step of the arc to be drawn; means for applying the extracted values to modify the temporarily registered values respectively of the same axis; and means for controlling repeated sequential operations of the accumulating means, the extracting means and the applying means to specify succeeding steps of arc drawmg.

2. Apparatus as claimed in claim 1, further including means for controlling the arithmetic sign of the values from the temporary registering means in dependence upon the particular quadrant with respect to the axes in which lies the arc to be drawn.

3. Apparatus as claimed in claim 2 including a cathode ray tube having a display face on which the arc is drawn by displacement of an electron beam and means for displacing the beam in X- and Y-axes respectively, irll which said means for applying the extracted values to specify the arc step includes X- and Y-position registers, the beam displacing means being responsive to values in the position registers respectively to position the beam relative to said display face; and means for adding into the position registers respectively said extracted values.

4. Apparatus as claimed in claim 3, in which said applying means for modifying the temporarily registered values includes summing means interposed between the respective accumulating means and the temporary register of each axis, the summing means being responsive to the value from the temporary register and the extracted value for that axis to enter the sum of these values into the temporary register.

5. Apparatus for controlling drawing of arcuate lines on a display face of a cathode ray tube by displacement of an electron beam, including for one of a pair of coordinate axes means for deriving a value representing the product of the radius of the required arc and the sine of a displacement angle 6 through which the radius is to be moved by a predetermined incremental distance along the arc to be drawn and for the other of the pair of the co-ordinate axes means for deriving a value representing the product of the radius and the cosine of the displacement angle, each value of the pair being expressed in terms of said incremental distance in binary code having n denominations; means for temporarily registering each value of the pair independently; accumulating means for each axis respectively, each accumulating means having input lines; first and second gating means for each axis, said first and second gating means both receiving the value from the temporary registering means for that axis and each having a group of n output lines, the output lines of the first gating means of an axis being connected to the input lines of the accumulating means of the same axis in denominational order with an effective n+1 denominational shift and the output lines of the second gating means being connected to the input lines of the accumulating means of the other axis in denominational order with an effective n/2 denominational shift; means for extracting from the accumulating means for each axis respectively those values from the n/2 most significant denominations; X- and Y- position registers, one for each axis respectively; beam displacement means for each axis responsive respectively to values in the X- and Y- position registers to position the cathode ray tube beam relative to the display face; means for adding the values extracted from the accumulating means into the X- and Y-position registers respectively; summing means interposed between the respective accumulating means and the temporary registering means for each axis, the summing means being responsive to the value from the temporary registering means and the value extracted from the accumulating means to enter the sum of those values into the temporary registering means thereby modifying the temporarily registered values; means for controlling repeated sequential operations of the accumulating means, the extracting means and the adding and summing means to specify succeeding steps of arc drawing and means for controlling the arithmetic sign of values from the temporary registering means in dependence upon the particular quadrant with respect to the axes in which lies the arc to be drawn.

6. Apparatus as claimed in claim 5 in which the means for controlling the arithmetic sign of values to be accumulated includes means for selectively forming the complement of an applied value, a complementing means being interposed between the temporary registering means and the gating arrangements of each axis independently.

7. Apparatus as claimed in claim 6 including means for indicating the completion of drawing of a quadrant of the arc, the controlling means being responsive to an indication from the indicating means to determine the selective operation of the complementing means.

8. Apparatus as claimed in claim 7 including means for registering a radius value representing the arc radius, the indicating means comparing the radius value with successive ones of said extracted values to produce a signal indicating the completion of a quadrant if the compared values differ by less than a predetermined value.

9. Apparatus as claimed in claim 8 including a further register interposed between the accumulating means and the position register for each axis, the indicating means being connected between the radius value register and said further register, the further register being arranged to receive the extracted values. 

1. Apparatus for controlling drawing of arcuate lines on a display face, including means for initially deriving a pair of values representing for co-ordinate axes respectively the product of the radius of the required arc and the sine of a displacement angle theta through which the radius is to be moved by a predetermined incremental distance along the arc to be drawn and the product of the radius and the cosine of the displacement angle theta , the pair of values being expressed in terms of said incremental distance in binary code having n denominations; means for temporarily registering the pair of values in each axis independently; accumulating means for each axis respectively; means for entering values from the temporary registering means into the accumulating means to accumulate independently for each axis respectively the temporarily registered value for the same axis right shifted by n + 1 denominations and the temporarily registered value for the other axis right shifted by n/2 denominations; means for extracting from the accumulating means for each axis respectively those values from the n/2 most significant denominations; means for applying the extracted values respectively to specify for the co-ordinate axes a step of the arc to be drawn; means for applying the extracted values to modify the temporarily registered values respectively of the same axis; and means for controlling repeated sequential operations of the accumulating means, the extracting means and the applying means to specify succeeding steps of arc drawing.
 2. Apparatus as claimed in claim 1, further including means for controlling the arithmetic sign of the values from the temporary registering means in dependence upon the particular quadrant with respect to the axes in which lies the arc to be drawn.
 3. Apparatus as claimed in claim 2 including a cathode ray tube having a display face on which the arc is drawn by displacement of an electron beam and means for displacing the beam in X- and Y-axes respectively, in which said means for applying the extracted values to specify the arc step includes X- and Y-position registers, the beam displacing means being responsive to values in the position registers respectively to position the beam relative to said display face; and means for adding into the position registers respectively said extracted values.
 4. Apparatus as claimed in claim 3, in which said applying means for modifying the temporarily registered values includes summing means interposed between the respective accumulating means and the temporary register of each axis, the summing means being responsive to the value from the temporary register and the extracted value for that axis to enter the sum of these values into the temporary register.
 5. Apparatus for controlling drawing of arcuate lines on a display face of a cathode ray tube by displacement of an electron beam, including for one of a pair of coordinate axes means for deriving a value representing the product of the radius of the required arc and the sine of a displacement angle theta through which the radius is to be moved by a predetermined incremental distance along the arc to be drawn and for the other of the pair of the co-ordinate axes means for deriving a value representing the product of the radius and the cosine of the displacement angle, each value of the pair being expressed in terms of said incremental distance in binary code having n denominations; means for temporarily registering each value of the pair independently; accumulating means for each axis respectively, each accumulating means having input lines; first and second gating means for each axis, said first and second gating means both receiving the value from the temporary regisTering means for that axis and each having a group of n output lines, the output lines of the first gating means of an axis being connected to the input lines of the accumulating means of the same axis in denominational order with an effective n+1 denominational shift and the output lines of the second gating means being connected to the input lines of the accumulating means of the other axis in denominational order with an effective n/2 denominational shift; means for extracting from the accumulating means for each axis respectively those values from the n/2 most significant denominations; X- and Y- position registers, one for each axis respectively; beam displacement means for each axis responsive respectively to values in the X-and Y- position registers to position the cathode ray tube beam relative to the display face; means for adding the values extracted from the accumulating means into the X- and Y-position registers respectively; summing means interposed between the respective accumulating means and the temporary registering means for each axis, the summing means being responsive to the value from the temporary registering means and the value extracted from the accumulating means to enter the sum of those values into the temporary registering means thereby modifying the temporarily registered values; means for controlling repeated sequential operations of the accumulating means, the extracting means and the adding and summing means to specify succeeding steps of arc drawing and means for controlling the arithmetic sign of values from the temporary registering means in dependence upon the particular quadrant with respect to the axes in which lies the arc to be drawn.
 6. Apparatus as claimed in claim 5 in which the means for controlling the arithmetic sign of values to be accumulated includes means for selectively forming the complement of an applied value, a complementing means being interposed between the temporary registering means and the gating arrangements of each axis independently.
 7. Apparatus as claimed in claim 6 including means for indicating the completion of drawing of a quadrant of the arc, the controlling means being responsive to an indication from the indicating means to determine the selective operation of the complementing means.
 8. Apparatus as claimed in claim 7 including means for registering a radius value representing the arc radius, the indicating means comparing the radius value with successive ones of said extracted values to produce a signal indicating the completion of a quadrant if the compared values differ by less than a predetermined value.
 9. Apparatus as claimed in claim 8 including a further register interposed between the accumulating means and the position register for each axis, the indicating means being connected between the radius value register and said further register, the further register being arranged to receive the extracted values. 